This command reduces the entries of an NCMatrix with respect to an NCGroebnerBasis.
i1 : A = QQ{x,y,z}
o1 = A
o1 : NCPolynomialRing
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i2 : f = y*z + z*y - x^2
2
o2 = zy+yz-x
o2 : A
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i3 : g = x*z + z*x - y^2
2
o3 = zx-y +xz
o3 : A
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i4 : h = z^2 - x*y - y*x
2
o4 = z -yx-xy
o4 : A
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i5 : I = ncIdeal {f,g,h}
2 2 2
o5 = Two-sided ideal {zy+yz-x , zx-y +xz, z -yx-xy}
o5 : NCIdeal
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i6 : Igb = ncGroebnerBasis I
--Calling Bergman for NCGB calculation.
--running: bergman -i /var/folders/46/9b86vqxj4hjcngvy7kd7sb140000gn/T/M2-12323-0/0.init -on-error exit --silent > /var/folders/46/9b86vqxj4hjcngvy7kd7sb140000gn/T/M2-12323-0/3.ter ... Complete!
2 2 2
o6 = y x-xy ; Lead Term = (y x, 1)
2 2 2
yx -x y; Lead Term = (yx , 1)
2
zx-y +xz; Lead Term = (zx, 1)
2
zy+yz-x ; Lead Term = (zy, 1)
2 2
z -yx-xy; Lead Term = (z , 1)
o6 : NCGroebnerBasis
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i7 : M = ncMatrix {{x, y, z}}
o7 = | x y z |
o7 : NCMatrix
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i8 : sigma = ncMap(A,A,{y,z,x})
o8 = NCRingMap A <--- A
o8 : NCRingMap
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i9 : N = ncMatrix {{M},{sigma M}, {sigma sigma M}}
o9 = | x y z |
| |
| y z x |
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| z x y |
o9 : NCMatrix
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i10 : N3 = N^3
| 2 2 2 2 3 2 2 3 2 2 3 2 2 2 2 |
o10 = | z x+zyz+zxy+yzy+y x+yxz+xz +xy +x z y+zyx+zxz+yz +y +yx +xzx+xyz+x y z +zy +zx +yzx+y z+yxy+xzy+xyx+x z |
| |
| 2 2 3 2 2 3 2 2 2 2 2 2 2 2 3 |
| z y+zyx+zxz+yz +y +yx +xzx+xyz+x y z +zy +zx +yzx+y z+yxy+xzy+xyx+x z z x+zyz+zxy+yzy+y x+yxz+xz +xy +x |
| |
| 3 2 2 2 2 2 2 2 2 3 2 2 3 2 2 |
| z +zy +zx +yzx+y z+yxy+xzy+xyx+x z z x+zyz+zxy+yzy+y x+yxz+xz +xy +x z y+zyx+zxz+yz +y +yx +xzx+xyz+x y |
o10 : NCMatrix
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i11 : N3red = N3 % Igb
| 2 3 2 2 2 2 2 2 2 3 2 3 2 3 |
o11 = | -y z+y +yxz-yxy+xyz+xy +2xyx+x z+3x y y z+yxz+2yxy+xyz+3xy -xyx-x z+x y+x 2y z+y +yxy+xyx+2x z+x |
| |
| 2 2 2 2 3 2 3 2 3 2 3 2 2 2 |
| y z+yxz+2yxy+xyz+3xy -xyx-x z+x y+x 2y z+y +yxy+xyx+2x z+x -y z+y +yxz-yxy+xyz+xy +2xyx+x z+3x y |
| |
| 2 3 2 3 2 3 2 2 2 2 2 2 2 3 |
| 2y z+y +yxy+xyx+2x z+x -y z+y +yxz-yxy+xyz+xy +2xyx+x z+3x y y z+yxz+2yxy+xyz+3xy -xyx-x z+x y+x |
o11 : NCMatrix
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